They were introduced to Britain by the Romans, are hated as pests and celebrated in children's books. Britain's estimated 40m rabbits cost the economy more than £260m a year including damage to crops, businesses and infrastructure, a report says today.

Their near 2,000-year survival as a non-native species makes them the costliest natural invader, according to research for the English, Scottish and Welsh governments by CABI, the international agriculture and environment organisation.

Japanese knotweed, introduced in the mid-19th century as an ornamental garden plant, has had far less time to rack up its own costly threats to the economy as it colonises roadsides, riverbanks and derelict land.

The total cost of such invasive species now reaches £1.7bn a year, mostly to the English economy, more than £250m in Scotland and £133m in Wales.

But the report also provides justification for the high costs of controlling non-natives. It is estimated that the current eradication programme for water primrose, a group of South American aquatic weeds which grow rapidly and can block waterways, will cost £73,000 – significantly less than the estimated £242m it would cost if the weeds were to become widely established as they have in in countries including France and Belgium.

Eradicating the grey squirrel, the North American 19th-century import which has put the native red in peril, may no longer be possible, the report concedes. An attempt to remove the species from Anglesey alone has so far cost £440,000. A conservative estimate puts the cost of a similar programme across Britain at £850m.

Even that would be relatively small if the Asian long-horned beetle, a wood-boring menace from Japan, Korea and China, and now threatening North American trees, became established in this country. The report suggests it would cost over £1.3bn to remove it from hardwood forests alone, let alone clear it from parks, gardens and hedgerows.

Richard Benyon, minister for the natural environment, said: "Invasive non-native species have a significant impact on the British economy and damage our own wildlife. The costs of controlling these species will rise unless society takes steps to prevent them taking hold and spreading."

"It becomes increasingly difficult and costly to control invasive non-native species as they become more established. Taking early action may seem expensive, but this report shows that it is the most effective approach, saving money in the long run and helping our native wildlife to thrive."

Roseanna Cunningham, Scottish minister for environment and climate change, said: "A better understanding of the negative impacts of invasive non-native species can help us raise awareness to help prevent introductions in the first place and to better respond to problems."

Top 10 economic menaces

Rabbits £263m a year

Japanese knotweed £179m

Common field speedwell/wild oat £100m

Rat £62m

Potato cyst nematode £50m

Non-native deer £35m

Varroa mite (a parasite of bees) £27m

Floating pennywort £25m

House mouse £18m

Grey squirrel £14m

• This article was amended on 15 December 2010. The original referred to the near 1,000-year survival of rabbits as a non-native species. This has been corrected.